Betting trip game

ABSTRACT

A method to operate non-pari-mutuel game of chance using a map of sites as playing surface with moving pieces called movers. The game rules a plurality of movements. A random draw of ruled movements moves movers from sites to sites, one round after another. Calculation of any possible movement outcome probabilities is provided. Players can bet on any one or more possible movement outcomes. The holder of a hanging bet earns credit to place free make-up bets. House edge functions with outcome probability as variable are suggested. An automatic computer/video version of the game is included.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is related to the application 691,944, filed on 1996Aug. 5, now U.S. Pat. No. 5,795,226, granted on 1998 Aug. 18. Theinventor's name was misprinted as Chen Yi. A certificate of correctionwas issued on 1998 Nov. 24. This application is related to applicationSer. No. 12/589,989 filed on 2009 Nov. 2, now abandoned, which is acontinuation application of Ser. No. 11/299,050 filed on 2005 Dec. 12,also abandoned.

BACKGROUND OF THE INVENTION

1. Field of Invention

This invention relates to games of chance, more specifically, to methodsof playing a betting game determined by one or multiple rounds of randomnumbers, managed by computer data possessing.

2. Prior Art

As far as playing surface is concerned every game with a plurality ofmoving pieces is prior art. As far as betting is concerned, anynon-para-mutual game of chance, such as craps, keno, roulette is priorart. As far as technology is concerned, games requiring bet slips andcomputer data processing such as racetrack operation or those listed inthe Information Disclosure Statement are all in a sense prior art.

3. Objects and Advantages

At the time of U.S. Pat. No. 5,795,226 being granted, I realized thatthe non-automatic version requires a 8′ by 8′ table, a rolling dice box,and so on, all made-to-order only. Its operation requires severalworkers. All this means high cost, which will result in high house edge,something I hate. Why not replace the big table by a monitor display?Why not let a keno bowl of balls to generate random numbers? Why notallow players to determine own track length, and start a race anytime?etc., various ideas of improvement. Besides, why not change the racingcharacteristic to movement from one site to another? Why not, instead ofracers finishing order, bet that Ann will make a trip from London toParis followed by Bob from Beijing to Tokyo and then to Sydney. So thisinvention originated, first disclosed in Ser. No. 11/299,050, thenamended in Ser. No. 12/589,989 with one more drawing, and now describedhere in further easy to understand detail.

This invention provides a keno-like game overcoming the followingweaknesses of today's keno: 1. Too low winning probabilities and no lowhouse edges. The highest keno winning probability is 1 in 4 while houseedge is at least 25% (an exception will be given below). Most casinogames have higher probabilities such as roulette “black-red”. House edgecan be less than 1% such as craps “free odds”. Looking into statistic onannual casino revenue and state lottery ticket sale, we see that thecommon gambling intention is inclined by far to catching low house edgeand more probable wins rather than becoming a millionaire fast. 2. Tootroublesome to place desired amount of bets. There are 3.5 quintillionpossible combinations of 20 out of 80. But, say, you want to play allpossible 6-spot catch-all combinations with numbers 1, 2, 3 plus anythree numbers from 4 to 80, you need to mark at least a few hundred betslips for your 73150 bets. No keno writer will be ready to help you.Besides, spots of a group of numbers are not adjacent to each other, orthere are more than two groups to circle, confusion will likely occur incomputerized digital scanning. 3. Every bet is determined by one draw.The only known exception is ‘Exacta’ occurred at Gold Coast, Las Vegas,which allowed players to mark the same number of spots, from one to ten,in two consecutive games, paying $1 per game plus $0.25 for exacta. Forthe best payoff is thus to mark one spot in each game which pay $3 forfirst game, $3 for second game and $4 for exacta, implying house edgesto be 25%, 25% and 0% respectively. Unfortunately, due to 0% on 25 centsand 25% on $2, there is no way to take advantage of a hanging ticket byplacing make-up bets after the first game. 4. Problematic to generaterandom numbers. Keno usually uses a bowl of 80 whirling balls to pushone at a time by air force into a selection tube until 20 are collected.The problem is, within a short period of action, not every ball canreach an appropriate position to be pushed. Besides, it occurred that acustomer remarked to a keno manager at a Las Vegas casino that Number 29never came up. Indeed, Number 29 was not in the cage. There will be nosuch problem if the game requires, say, to pick just one number out ofsix while the cage contains 24 balls, four copies of each number.

In early 1930s the Liberty Bell slot machines with 3-reel, 10 symbolsper reel were installed in Las Vegas casinos for the purpose of keepingwives and girlfriends entertained while serious gamblers played atgaming tables. They became one-armed-bandits indicating that payoffswere very poor. Then the computer technology turned them into slot/videomonsters, and since 1990s the biggest money-makers in casinos. Theircommon characteristics are:

(A′) Every player occupies one individual machine throughout the wholeplaying period.

(B′) Most machines are bulky, mesmerizing with video displays, high-techsound effects for entertainment purpose only. Each costs thousands ofdollars to manufacture and transport.

(C′) Physical/simulated wheels/reels.

(D′) Hidden virtual wheels/reel to produce outcomes technically known tothe operator only.

(E′) RNG software to ensure operator's maximal profit allowed bygovernment gaming regulations.

(F′) Regardless of millions of possible outcomes, only limited numberavailable to bet on.

(G′) Besides limited number of paylines, there are “multi-line”,“bonus-round”, “option-buy”, “scatter-pay”, “progressive” etc., justluring less intelligent people to wager more and hard to get up from themachine with a possibility of money left to be won. But there is noinformation materials about which, when and how the possibility mayoccur.

Due to (D′) to (G′), they will not be touched by serious orsophisticated players, who have been simply ignored by the gamingindustry.

This invention is to disclose a betting movement operation with thefollowing advantages:

(A) No physical movers or playing surfaces or simulated reels/wheels.—thus convenient and low-cost.

(B) Movers and playing surfaces are displayed on monitors and printed onbet slips/tickets—thus convenient and low-cost.

(BB) Anywhere electronically connected to the game control center canallow placing bets by computerized pointer clicking or screentouching—thus convenient and low-cost.

(C) Using simple random draws to determine the movements of all moverson any bet tickets—thus convenient and low-cost.

(D) No RNG software—thus no unknown bias against players.

(E) No virtual reels/wheels—thus assuring no hidden outcomes.

(F) Players don't need to stay in front of individual machines to watchoutcomes—thus convenient and low-cost.

(G) Players can always arbitrarily bet on any amount of billionspossible outcomes with winning probabilities ranging from 99% to abillionth—thus by far more attractive than limited number of paylines.

(GG) Operators can avoid huge payoff by limiting bet amount.

(H) Every outcome with disclosed mathematical probability—thusattractive to serious players.

(I) Players are assured that the game operator's only advantage is houseedge.

(J) Based on known probabilities and known house edges, player canfigure out playing plans in advance—thus attractive to serious players.

(K) There are multi-draw bets resulting in hanging bets allowing holderto earns non-cashable credit to place free make-up credit bets—thusattractive to serious players.

(KK1) Credit betting allows player to give up a less probable bigger winfor a sure smaller win scientifically—thus attractive to reasonableplayers.

(KK2) Credit betting reduces the operator's risk of a sudden hugepayoff.

(L) House edges formulae for all type of bets, with or without credit,based on final total winning probability, to be applied only topayoff—thus attracting players to place multi-draw bets together withmake-up bets, using credit or not; consequently, having more fun andlosing money less fast, while the cost of computerized handling isimmaterial.

(M) Playing surfaces and ruled movements were unknown to the public till2005 Dec. 12.

(MM) There is no known prior art requiring specific modular arithmeticrequired and presented here, which may mesmerize curious intelligentpeople.

(N) Placing a bet may raise the feeling of going on a trip with lovedones.

Here is a very simple example of dealing with hanging bet for a surewin. Say, I purchase a $100 2-Draw ticket on Ann to go to London firstand then to Paris, each time one of seven possible destinations. Whenthe first draw indeed moves Ann to London, I receive no payoff butcredit $700 for the hanging bet. To take advantage of it, I can use,say, 60% credit to bet $70 on each of six destinations other than Paris.The result will be a payoff of either $40*49*94.13%=$1845.01, in case ofParis, or $70*7*94.13%=$461.25, in any other cases, where house edgesare all 5.87% based on probability 1/49. Now, how about I don't want touse any credit? Then, in order to be a sure winner, I need $3,600 toplace six $600 make-up bets on all destinations other than Paris. Theresult will be either a payoff of $100*49*94.13%=$4612.51 on my originalticket where house edge is 5.87% based on probability 1/49, or a payoffof $600*7*95.75%=$4025.10 on any one of six make-up bets where houseedge is 4.25% based on probability 1/7. Anyway, I can turn the hangingbet into a sure winner.

More numerical examples to see advantages (G) to (L) will be givenbelow.

SUMMARY OF THE INVENTION

The invention provides a game of chance with a playing surface. Theplaying surface is a map of sites on which there are movers, eachdenoted by a circled number. One or more TV/computer monitors will berequired to display the playing surface with movers.

The invention provides a plurality of ruled movements directing moversto move from one site to another. A random draw device functioning likethe one used at keno will be required to draw ruled movements at random.Here we use one cage for each mover in which there are equally many,say, four balls for each ruled movement.

In case of non-automatic version, the game requires a player calledoperator to conduct. Other players are bettors. The operator executesrandom draw of ruled movements, one round after another, moving eachmover accordingly once per round. Henceforth every round of drawing willbe called a ‘Draw’.

The invention provides a plurality of paper bet slips showing either thewhole playing surface with all movers or a regional playing surface,called moving region, for each bet-on mover. As at a racetrack bettorsmark paper bet slips to place bets. The bettor can select one or severalmovers together with one or several sites which the bettor expects tomatch the outcomes of upcoming one or several draws.

The game requires wagering machines, similar to those used atracetracks, connected to a data processing computer. They examine markedbet slips and print bet tickets showing all officially accepted bets,The computer records and processes betting data with drawing results.Wagering machines may also display playing surfaces to allow bettors toplace bets without paper bet slips like at racetrack.

A multi-Draw bet becomes or remains ‘hanging’ if it contains a selectionof movers with movements matching the last draw outcomes, and thus has achance to be a winner later on. The invention provides the option thatany hanging bet holder earns credit, which can be used fully or in partto place free make-up bets, henceforth called credit bets.

The invention provides formulae to calculate winning probability ofevery bet as well as all payoffs and credits.

The invention provides house edge formulae to be applied to payoffs.

In case of automatic version, a video/computer devise will be required,and a player conducts draw and places bets all by oneself.

DRAWINGS

In the figures, like reference numerals will refer to like elementsthroughout. For example, numeral 12 will be applied to any mover in anysite, regardless whether the mover is an ‘actual’ one displayed on amonitor, or ‘selectable’ one on a bet slip or bet ticket. But there aredifferent numerals for sites according to Draw #.

FIG. 1 is a flowchart illustrating the process of gaming operation.

FIG. 2X illustrates a playing surface containing seven sites 11 and sixmovers 12 together with all reference numerals and lead lines. So manynumerals and lead lines make it hardly acceptable.

FIG. 2 illustrates a playing surface identical to that of FIG. 2X.However, all but one reference numeral 11 and its lead line will beomitted, and all but one reference numeral 12 and its lead line will beomitted.

The omission stated above should be easily understandable to everyone.It is not only for aesthetic sake, but also necessary wherever hardlypossible to design an acceptable drawing of numerous identical referencenumerals with their lead lines. Thus, this kind of omission of identicalreference numerals with their lead lines will occur in most of thefollowing figures.

Besides, in the figures there are icons/spots named ‘Simple’, ‘Site’,‘Mixed’, ‘Chain’, ‘Credit’ and ‘New slip’, all without referencenumerals due to no practical necessity.

Besides, there will be no practical need of reference numerals infigures of non-chain-bet ticket, while in chain-bet tickets, referencenumerals just provided for placing credit bets.

Besides, in all figures of bet tickets:

Sign “X” indicates a selection made before Draw 1.

Sign “=” indicates a selection made after Draw 1.

A gray site indicates where a mover locates before a Draw.

A gray mover indicates where it locates after a Draw.

FIG. 3 illustrates a playing surface containing ten sites and sixmovers. Just as in FIG. 2, all but one reference numeral 11 with leadline and one numeral 12 with lead line will be omitted.

FIG. 3′ illustrates a display to show movers locating in gray sitesbefore Draw and becoming gray after Draw. It is a detailed record withno need of reference numerals for further explanation or reference.

FIG. 4 illustrates a playing surface containing twenty sites and sixmovers. All but one reference numeral 11 with lead line and one numeral12 with lead line will be omitted.

FIG. 4′ illustrates a display to show movers locating in gray sitesbefore Draw and becoming gray after Draw. It is a detailed record withno need of reference numerals for further explanation or reference.

FIG. 2A is a blank bet slip using playing surface as shown in FIG. 2.Without omission there will be six selectable movers 12 in each of sevensites 31, seven sites 41, seven sites 51, and seven sites 61. All butone reference numeral 31, one 41, one 51, one 61 with lead line will beomitted, and all but one reference numeral 12 with lead line related toeach of those sites will be omitted. Without omission there will be sixselectable amounts per bet 70. All but one numeral 70 with lead linewill be omitted. Without omission there will be six total bet amounts80. All but one numeral 80 with lead line will be omitted.

FIGS. 2B, 3B and 4B are each a ‘simple’ bet ticket showing all bet-onmovers in selected sites and wagering amounts based on selections markedon a blank bet slip as shown in FIGS. 2A, 3A and 4A respectively. Eachof them is a detailed record with no need of reference numerals forfurther explanation or reference.

FIGS. 2C, 3C and 4C are each a ‘site’ bet ticket showing all bet-onmovers in selected sites and wagering amounts based on selections markedon a blank bet slip as shown in FIGS. 2A, 3A and 4A respectively. Eachof them is a detailed record with no need of reference numerals forfurther explanation or reference.

FIGS. 2D, 3D and 4D are each a ‘mixed’ bet ticket showing all bet-onmovers in selected sites and wagering amounts based on selections markedon a blank bet slip as shown in FIGS. 2A, 3A and 4A respectively. Eachof them is a detailed record with no need of reference numerals forfurther explanation or reference.

FIG. 2E is a 4-Draw ‘chain’ bet ticket showing all betting data based onselections marked on a blank bet slip as shown in FIG. 2A. It shows allbet-on movers marked with “X” in selected sites. It is also a slip forcredit bets. There are selectable unmarked movers 12 in some of sevensites 41, seven sites 51, and seven sites 61. All but one referencenumeral 41, one 51, one 61 with lead line will be omitted, and all butone reference numeral 12 related to each of those sites with lead linewill be omitted. Without omission there will be twenty-seven selectablecredit percentages 99. All but one of them with lead line will beomitted.

FIG. 2F is the bet ticket as shown in FIG. 2E becoming hanging and beingrevised. It shows selected credit percentage and Draw 2 credit betselections marked by “=”. It also shows Draw 1 outcomes by gray movers.Since it can be used as slip for further credit bets, there areselectable unmarked movers 12 in sites 51 and sites 61. All but onereference numeral 51 and one 61 with lead lines will be omitted, and allbut one reference numeral 12 related to each of those sites with leadline will be omitted. Without omission there will be eighteen selectablecredit percentages 99. All but one of them with lead line will beomitted.

FIG. 2G is the bet ticket as shown in FIG. 2F remaining hanging andbeing revised. It shows selected credit percentage and Draw 3 credit betselections marked by “=”. It also shows Draw 2 outcomes by gray movers.Since it can be used as slip for further credit bets, there areselectable unmarked movers 12 in sites 61. All but one reference numeral61 with lead line and one numeral 12 with lead line will be omitted.Without omission there will be nine selectable credit percentages 99.All but one of them with lead line will be omitted.

FIG. 2H is the bet ticket as shown in FIG. 2G remaining hanging andbeing revised. It shows selected credit percentage and Draw 4 credit betselections marked by “=”. It also shows Draw 3 outcomes by gray movers.

FIGS. 3A and 4A are blank bet slips using playing surfaces as shown inFIGS. 3 and 4 respectively. Omissions of reference numbers and leadlines occur similarly to that in FIG. 2A.

FIG. 3AA is a blank chain bet slips using playing surface as shown inFIG. 3. Without omission there will be six moving regions 30, six 40,six 50 and six 60. All but one numeral 30, one 40, one 50, and one 60with lead line will be omitted. There are selectable unmarked movers 12in some of thirty-six sites 31, thirty-six sites 41, thirty-six sites51, and thirty six sites 61. All but one reference numeral 31, one 41,one 51 and one 61 with lead lines will be omitted, and, related to eachof those sites, all but one numeral 12 with lead line will be omitted.Without omission there will be six selectable amounts per bet 70. Allbut one numeral 70 with lead line will be omitted. Without omissionthere will be six selectable total bet amounts 80. All but one numeral80 with lead line will be omitted.

FIG. 3E is a 3-Draw bet ticket showing all betting data based onselections marked on a blank bet slip as shown in FIG. 3AA. It shows allbet-on movers marked with “X” in selected sites. It is also a slip forcredit bets. There are selectable unmarked movers 12 in some of thirtysites 41, and thirty sites 51. All but one reference numeral 41, one 51with lead line will be omitted, and, related to each of those sites, allbut one reference numeral 12 with lead line will be omitted. Withoutomission there will be eighteen selectable credit percentages 99. Allbut one of them with lead line will be omitted.

FIG. 3F is the bet ticket as shown in FIG. 3E becoming hanging and beingrevised. It shows selected credit percentage and Draw 2 credit betselections marked by “=”. It also shows Draw 1 outcomes by gray movers.Since it can be used as slip for further credit bets, there areselectable unmarked movers 12 in sites 51. All but one reference numeral51 with lead line will be omitted. Without omission there will be nineselectable credit percentage 99. All but one of them with lead line willbe omitted.

FIG. 3G is the bet ticket as shown in FIG. 3F remaining hanging andbeing revised. It shows selected credit percentage and Draw 3 credit betselections marked by “=”. It also shows Draw 2 outcomes by gray movers.

FIG. 4AA is a blank chain bet slips using playing surface as shown inFIG. 4. Without omission there will be six moving regions 30, six 40,six 50 and six 60. All but one reference numeral 30, one 40, one 50, andone 60 with lead line will be omitted. There are selectable unmarkedmovers 12 in some of fifty-four sites 31, fifty-four sites 41, fiftyfour sites 51, and fifty four sites 61. All but one numeral 31, one 41,one 51 and one 61 with lead lines will be omitted, and, related to eachof those sites, all but one reference numeral 12 with lead line will beomitted. Without omission there will be six selectable amounts per bet70. All but one numeral 70 with lead line will be omitted. Withoutomission there will be six selectable total bet amounts 80. All but onenumeral 80 with lead line will be omitted.

FIG. 4E is a 3-Draw bet ticket showing all betting data based onselections marked on a blank bet slip as shown in FIG. 4AA. It shows allbet-on movers marked with “X”. It is also a slip for credit bets. Thereare selectable unmarked movers 12 in some of thirty-six sites 41 andthirty six sites 51. All but one reference numeral 41, one 51 with leadline will be omitted, and, related to each of those sites, all but onereference numeral 12 with lead line will be omitted. Without omissionthere will be eighteen selectable credit percentages 99. All but one ofthem with lead line will be omitted.

FIG. 4F is the bet ticket as shown in FIG. 4E becoming hanging and beingrevised. It shows selected credit percentage and Draw 2 credit betselections marked by “=”. It also shows Draw 1 outcomes by gray movers.Since it can be used as slip for further credit bets, there areselectable unmarked movers 12 in some of thirty-six sites 51. All butone reference numeral 51 with lead line will be omitted. Withoutomission there will be nine selectable credit percentage 99. All but oneof them with lead line will be omitted.

FIG. 4G is the bet ticket as shown in FIG. 4F remaining hanging andbeing revised. It shows selected credit percentage and Draw 3 credit betselections marked by “=”. It also shows Draw 2 outcomes by gray movers.

FIG. 5 shows a betting activity statement.

FIG. 6 is a line graph to show house edge formula based on winningprobability.

Description of Various Playing Surfaces with Ruled Movements

Playing surfaces 10 in FIGS. 2, 3 and 4 will be displayed on a monitorto indicate the locations of movers either set arbitrarily by the gameoperator before the start of any betting action or determined by thelast draw of movements.

A ruled movement aiming at a specific location will be called ‘jump’;otherwise, ‘non-jump’. All movements on playing surface 10 as shown inFIG. 2 are ‘jump’, while those on playing surface 10 as shown in FIGS. 3and 4 are ‘non-jump’. As defined below, on any given playing surface,for a mover in any location, there is always the same number w ofpossible ruled movements.

Playing surface 10 in FIG. 2 contains seven sites 11, on each there aresix movers 12.

There are w=7 ruled movements for this playing surface, denoted ‘A’,‘B”, ‘C’, ‘D’, ‘E’, ‘F’, and ‘G’, as defined below:

-   -   ‘A’ moves the concerning mover to ‘A’, inclusive from ‘A’.    -   ‘B’ moves the concerning mover to ‘B’, inclusive from ‘B’.    -   ‘C’ moves the concerning mover to ‘C’, inclusive from ‘C’.    -   ‘D’ moves the concerning mover to ‘D’, inclusive from ‘D’.    -   ‘E’ moves the concerning mover to ‘E’, inclusive from ‘E’.    -   ‘F’ moves the concerning mover to ‘F”, inclusive from ‘F’.    -   ‘G’ moves the concerning mover to ‘G’, inclusive from G’.

Playing surface 10 in FIG. 3 contains ten sites 11, on each there aresix movers 12. Similarly to most computer/video games, it is necessaryto regard the top border line 1001 as identical to the bottom line 1002.Due to this identification, the surface is suitable to be painted on acylinder, and thus henceforth to be referred as ‘cylinder’. There are‘A’ and ‘L’ painted outside playing surface 10 shown in FIG. 3 tovisualizes this crossing border down/up situation. That is to tell: site‘A’ lies one site downward to site ‘L’, two sites downward to site ‘K’,site ‘B’ lies two sites downward to site ‘L’; while site ‘L’ lies onesite upward to site ‘A’, two sites upward to site ‘B’ and three sitesupward to site ‘C’, site ‘K’ lies two sites upward to site ‘A’ and threesites upward to site ‘B’, site ‘H’ lies three sites upward to site ‘A’.

There are w=6 ruled movements for this playing surface, denoted ‘00’,‘U1’, ‘U2’, ‘U3’, ‘D1’, and ‘D2’, as defined below:

-   -   ‘00’ keeps the concerning mover unmoved.    -   ‘U1’ moves the concerning mover one site upward.    -   ‘U2’ moves the concerning mover two sites upward.    -   ‘U3’ moves the concerning mover three sites upward.    -   ‘D1’ moves the concerning mover one site downward.    -   ‘D2’ moves the concerning mover two sites downward.

Playing surface 10 in FIG. 4 contains twenty sites 11, on each there aresix movers 12. Here it is necessary to regard the top border line 1001as identical to the bottom one 1002, the left border line 1003 identicalto the right one 1004. Due to this identification, the surface issuitable to be painted on a torus, and thus henceforth will be referredto as ‘torus’. There are sites ‘DE’, ‘DA’ etc. painted outside playingsurface 100 to visualizes this crossing border situation. That is totell: site ‘AA’ lies surrounded by site ‘DA’ in the north, by site ‘DB’in the northeast, by site ‘DE’ in the northwest, by site ‘AB’ in theeast, by site ‘AE’ in the west, by site ‘BA’ in the south, by site ‘BB’in the southeast, and by site ‘BE’ in the southwest; while site ‘AB’lies surrounded by site ‘DB’ in the north, by site ‘DC’ in thenortheast, by site ‘DA’ in the northwest, by site ‘AC’ in the east, bysite ‘AA’ in the west, by site ‘BB’ in the south, by site ‘BC’ in thesoutheast, and by site ‘BA’ in the southwest; and so on similarly ascompletely shown in FIG. 4.

There are w=9 ruled movements for this playing surface, denoted ‘00’‘N’, ‘E’, ‘W’, ‘S’, ‘NE”, ‘NW’, ‘SE’, and ‘SW’, as defined below:

-   -   ‘00’ keeps the concerning mover unmoved.    -   ‘N’ moves the concerning mover to the adjacent site lying north.    -   ‘E’ moves the concerning mover to the adjacent site lying east.    -   ‘W’ moves the concerning mover to the adjacent site lying west.    -   ‘S’ moves the concerning mover to the adjacent site lying south.    -   ‘NE’ moves the concerning mover to the adjacent site lying        northeast.    -   ‘NW’ moves the concerning mover to the adjacent site lying        northwest.    -   ‘SE’ moves the concerning mover to the adjacent site lying        southeast.    -   ‘SW’ moves the concerning mover to the adjacent site lying        southwest.

Description of Placing Bets

There are Draw 1 to Draw 4 1-Draw bets, further classified as ‘simple’,‘site’ or ‘mixed’. There are multi-Draw or n-Draw ‘chain’ bets, wheren=2 to 4, further classified as 2-Draw, 3-Draw or 4-Draw. All bets madeon one bet slip are of the same class/type. Bet slips can be printed onpaper as well as displayed on screen of a wagering machine.

On bet slips as shown in FIGS. 3A, 3AA, 4A and 4AA there is area 100 forbettors, just for reference sake, optionally to mark movers' locationsas displayed on the monitor.

On every bet slip there is a naming area 110 just intended to enhancefun unrelated to game rules. Here is nothing for a bettor to mark.

On every bet slip there is a spot ‘credit’. It will be marked only by abettor using a credit voucher as fund as explained later on.

The bet slip as shown in FIG. 2A will be used for any type bet usingplaying surface as shown in FIG. 2. The bet slip as shown in FIG. 3Awill be used for 1-Draw bets using playing surface as shown in FIG. 3,The bet slip as shown in FIG. 3AA will be used for n-Draw bets usingplaying surface as shown in FIG. 3. The bet slip as shown in FIG. 4Awill be used for 1-Draw bets using playing surface as shown in FIG. 4,The bet slip as shown in FIG. 4AA will be used for n-Draw bets usingplaying surface as shown in FIG. 4. Besides bets using bet slips,‘credit’ bets can be placed on a bet ticket as explained later on. Amover once selected in a Draw will be referred to as a ‘bet-on’ mover ofthat Draw. A site in which a bet-on mover is located will be referred toas ‘selected’ site of that mover. Bet-on-mover-site is combining bet-onmover with selected set.

Using bet slip as shown in FIG. 2A, the bettor must mark to selectexactly one of ‘simple’, ‘site’, ‘mixed’ or ‘chain’. Using bet slip asshown in FIG. 3A or FIG. 4A, the bettor must mark to select exactly oneof ‘simple’, ‘site’ or ‘mixed’.

Regardless of using which bet slip, the bettor must mark to selectexclusively either one ‘amount per bet’ 70 or one ‘total bet amount’ 80for all bets marked on the slip.

Referring now to bet slip as shown in FIGS. 2A, 3A and 4A, there areareas Draw 1 to Draw 4 for bettors to mark bet-on movers 12 in sites 31in Draw 1, in sites 41 in Draw 2, in sites 51 in Draw 3, and in sites 61in Draw 4 respectively.

When the movement is ‘jump’, all sites 31 in Draw 1, 41 in Draw 2, 51 inDraw 3 and 61 in Draw 4 as shown in FIG. 2A are reachable by a singlemovement. When the movement is ‘non-jump’, referring to FIG. 3A or 4A,not every site 31 in Draw 1 is reachable by a single ruled movement. Amover in an unreachable site will be automatically cancelled by thecomputer, when the bet slip is submitted for approval. Regardless of‘jump’ or ‘non-jump’, all sites 41, 51 and 61 are all reachable by morethan one movement.

Referring now to FIGS. 3AA and 4AA, instead of a complete playingsurface for all movers, there are for each mover a moving region 30, 40,50 and 60. In each of them, the concerning mover is located in the graysite before the concerning Draw. And each of them contains exactly allsites reachable by a single ruled movement of the mover from the graysite. The bettor will mark to bet mover 12 moving to site 31 in Draw 1,to site 41 in Draw 2, to site 51 in Draw 3, to site 61 in Draw 4respectively. Thus, in multi-Draw, every movement becomes ‘jump’,namely, from the gray site jumping to any site within the concerningmoving region.

In the following, * is the multiplication operator, and ̂ the exponentoperator. For any f(M), Σ(f(M)) is summation of f(M) over all M to bespecified, and Π(f(M)) is multiplication of f(M) over all M to bespecified. Mathematically in general, M is a variable of function f,where f remains to be defined. Here, M is the numeral of a numberedmover. We define first f(M) to be #31(M), #41(M), #51(M) and #61(M)respectively as the number of selected sites 31, 41, 51 and 61 of bet-onmover M. We will later on define f(M) to be #d1(i(M)), p1M, pnM, etc.

To place 1-Draw bets, the bettor marks to select one or several movers12 in sites 31, 41, 51 and 61. Every selected 12 becomes a bet-on mover.The bettor can play any one or more Draws on one bet slip. All Draws areindependent. It is allowed, for example, to select some movers in sites31, some in sites 51, but none in 41 and 61 for playing Draw 1 and Draw3 only.

In the ‘simple’ case, every bet-on-mover-site counts a bet, orequivalently, every bet-on mover in each selected site counts a bet; orequivalently, every site with a bet-on mover count a bet. Thus, thenumbers of Draw 1, 2, 3, and 4 ‘simple’ bets are Σ(#31(M)), Σ(#41(M)),Σ(#51(M)), and Σ(#61(M)) respectively. For example, in FIG. 2B, Draw 1,there are bet-on mover M=#1 in sites C and E resulting in #31(#1)=2;bet-on mover M=#2 in sites A, E and G resulting in #31(#2)=3; bet-onmover M=#3 in sites A and D, resulting in #31(#3)=2; bet-on mover M=#4in site F only, resulting in #31(#4)=1; bet-on mover #5 in sites A, C,D, F and G, resulting in #31(#5)=5; and bet-on mover M=#6 in sites B, Dand G resulting in #31(#6)=3. Thus, Σ(#31(M))=2+3+2+1+5+3=16. The bettorwins, draw by draw independently, whenever there is one bet-on mover ina selected site 31 matching the outcomes of Draw 1; one bet-on mover ina selected site 41 matching the outcomes of Draw 2. one bet-on mover ina selected site 51 matching the outcomes of Draw 3; one bet-on mover ina selected site 61 matching the outcomes of Draw 4.

In the ‘site’ case, every selected site with all bet-on movers insidecounts a bet. Thus, the numbers of Draw 1, 2, 3, and 4 ‘site’ bets arenumbers of selected sites 31, 41, 51, and 61 respectively. The bettorwins, Draw by Draw independently, whenever there is a selected site 31with all bet-on movers in it matching the outcomes of Draw 1; a selectedsite 41 with all bet-on in it movers matching the outcomes of Draw 2; aselected site 51 with all bet-on movers in it matching the outcomes ofDraw 3; a selected site 61 with all bet-on movers in it matching theoutcomes of Draw 4. Note that the outcomes of non-bet-on movers have noeffect.

In the ‘mixed’ case, for any Draw, every distinct combination of eachbet-on mover and each selected site counts a bet. Thus, the numbers ofDraw 1, 2, 3, and 4 ‘mixed’ bets are Π(#31(M)), Π(#41(M)), Π(#51(M)),and Π(#61(M)) respectively. For example, in FIG. 2D, Draw 1, there arebet-on mover M=#2 in sites A, B, C, D, E and F resulting in #31(#2)=6;bet-on mover M=#3 in sites B, C, D, F and G resulting in #31(#3)=5;bet-on mover M=#4 in sites A, C, D, E and G resulting in #31(#4)=5;bet-on mover M=#5 in sites A, B, D, E, F and G resulting in #31(#5)=6.Thus, there are Π(#31(M))=6*5*5*6=900 distinct combinations making 900Draw1 ‘mixed’ bets. For example, in FIG. 2D, Draw 4, there are bet-onmover M=#1 in sites A, C, D, E, F and G resulting in #61(#1)=6; bet-onmover M=#3 in sites B, C, D and F resulting in #61(#3)=4; bet-on moverM=#4 in sites A, B, E and F resulting in #61(#4)=4; bet-on mover M=#6 insites A, B, D, E, F and G resulting in #61(#6)=6. Thus, there areΠ(#31(M))=6*4*4*6=576 distinct combinations making 576 Draw 4 ‘mixed’bets. The bettor wins, draw by draw independently, if there is acombination matching the outcomes. Since only one combination can matchthe outcomes, for each Draw, the ‘mixed’ bets on a ticket can bring inone winner only. For example, the bettor wins if the outcomes in FIG.2D, Draw 1, are #2 in A, #3 on G, #4 in C and #5 in B; or are #2 in D,#3 in C, #4 in D and #5 in A.

To place 2-Draw bets, the bettor marks to select first one or severalmovers in site 31 for Draw 1 just like making a Draw 1 ‘mixed’ bets, andthen one or several movers in sites 41 for Draw 2. Every Draw 1 bet-onmover must be bet-on in Draw 2 and vice versa. In the case of using betslip as shown in FIG. 3AA or FIG. 4AA, gray site 31 is where aconcerning mover is located before Draw 1: gray site 41 is where aconcerning mover is located before Draw 2, which is yet unknown. Thesame numbered bet-on mover in any selected site 41 is valid for allselected sites 31, no matter what the outcomes of Draw 1 may be. If thebettor wants a certain selected site 41 just for a certain selected site31, then it is necessary to use separate bet slips. For example, usingone bet slip you can bet a mover moves first either to east or west andthen either to north or south. Here are four bets on one slip. If youwant to bet that the mover moves either ‘first to east then to north’ or‘first to west then to south’, i.e., two bets only, then you need toplace them separately using two bet slips.

2-Draw bets are to combine Draw 1 ‘mixed’ bets with Draw 2 ‘mixed’ bets.Thus, the numbers of 2-Draw bets is Π(#31(M)*Π(#41(M)). For example, ifthere are bet-on movers A, B and C with #31(A)=4, #31(B)=3, #31(C)=5,#41(A)=2, #41(B)=6, and #41(C)=1, then the total number of bets isΠ(#31(M)*Π(#41(M))=(4*3*5)*(2*6*1)=720. If the ticket contains acombination matching the outcomes of Draw 1, then there will beΠ(#41(M)) ‘hanging’ bets. Then it wins if it contains a combinationmatching the outcomes of Draw 2. Since only one combination can matchthe outcomes, every slip can bring in one winner only.

To place 3-Draw bets the bettor marks to select first just as explainedin the 2-Draw case; then one or several movers in sites 51 for Draw 3.Every Draw 1 and Draw 2 bet-on mover must be bet-on in Draw 3 and viceversa. In the case of using bet slip as shown in FIG. 3AA or FIG. 4AA,gray site 51 is where a concerning mover is located before Draw 3, whichis yet unknown. The same numbered bet-on mover in any selected site 51is valid for all selected sites 31 and 41, no matter what the outcomesof Draw 1 and Draw 2 may be. If the bettor wants a certain selected site51 just for a certain selected sites 31 and 41, then it is necessary touse separate bet slips.

3-Draw bets are to combine 2-Draw bets with Draw 3 ‘mixed’ bets. Thus,the numbers of 3-Draw bets is Π(#31(M)*Π(#41(M))*Π(#51(M)). If theticket contains a combination matching the outcomes of Draw 1, thenthere will be Π(#41(M))*Π(#51(M)) ‘hanging’ bets. Next, if it contains acombination matching the outcomes of Draw 2, then there will beΠ(#51(M)) bets remaining hanging. Finally it wins if it contains acombination matching the outcomes of Draw 3. Since only one combinationcan match the outcomes, every slip can bring in one winner only.

To place 4-Draw bets the bettor marks to select first just as explainedin the 3-Draw case; then one or several movers in sites 61 for Draw 4.Every Draw 1 to Draw 3 bet-on mover must be bet-on in Draw 4 and viceversa. In the case of using bet slip as shown in FIG. 3AA or FIG. 4AA,gray site 61 is where a concerning mover is located before Draw 4, whichis yet unknown. The same numbered bet-on mover in any selected site 61is valid for all selected sites 31, 41, and 51, no matter what theoutcomes of Draw 1 to Draw 3 may be. If the bettor wants a certainselected site 61 just for a certain selected sites 31, 41, and 51, thenit is necessary to use separate bet slips.

4-Draw bets are to combine 3-Draw bets with Draw 4 ‘mixed’ bets. Thus,the numbers of 4-Draw bets is Π(#31(M)*Π(#41(M))*Π(#51(M))*Π(#61(M)). Ifthe ticket contains a combination matching the outcomes of Draw 1, thenthere will be Π(#41(M))*Π(#51(M)*Π(#61(M)) hanging bets. Next, if itcontains a combination matching the outcomes of Draw 2, then there willbe Π(#51(M))*Π(#61(M)) bets remaining hanging. Again, if it contains acombination matching the outcomes of Draw 3, then there will beΠ(#61(M)) bets remaining hanging Finally it wins if it contains acombination matching the outcomes of Draw 4. Since only one combinationcan match the outcomes, every slip can bring in one winner only.

Every marked bet slip will be checked and approved by the computer inorder to issue a bet ticket as shown in FIGS. 2B, 3B and 4B for ‘simple’bets: or FIGS. 2C, 3C and 4C for ‘site’ bets or FIGS. 2D, 3D and 4D for‘mixed’ bets; or FIGS. 2E, 3E and 4E for ‘chain’ bets. Each bet ticketshows an assigned number, and all data based on marked selections of abet slip, which includes type of bets, total number of bets, per betamount or total bet amount, and the concerning Draw # of Draw 1.Besides, except movements being ‘jump’, the bet ticket prints theplaying surface with start locations of all movers before Draw 1, whichare either the official results of the draw prior to Draw 1 or set bythe operator before the start. In the case of 1-Draw bets, only bet-onmovers in selected sites will show up in the bet ticket. In the case of‘chain’ bets, all bet-on movers in selected sites will be marked with“X”. Data shown in a bet ticket issued before the concerning Draw 1 willbe referred to as ‘original’. Data occurred after Draw 1 are ‘updated’.

After Draw 1, every hanging n-Draw bet earns credit—its amount to beshown later on—. Any ticket containing a hanging bet can be used as betslip to place credit bets as follows: The bettor marks to select ‘creditpercentage’ 99 and either ‘new slip’ or not. In case of no new bet slip,each existing bet-on-mover-site in Draw 2 on the bet ticketautomatically counts as a Draw 2 ‘simple’ bet, the bettor may mark toselect bet-on movers in additional sites 41 to bet on. The total hangingbet credit modified by selected percentage 99, referred to as r2, willbe evenly applied to all Σ(#41(M)) Draw 2 ‘simple’ bets. Every original$a bet will be reduced to a $a*(100−r2)% bet, The ticket as bet slipwill be approved by the computer so that a revised ticket as shown inFIG. 2F, 3F or 4F can be issued. In addition to existing data therevised ticket shows selected credit percentage for Draw 2, and all newbet-on selections marked with “=”. Besides, every bet-on mover will showup in gray to indicate its location determined by Draw 1. The bettorwith no intention to make credit bets may submit a hanging bet ticketwithout any credit bet selection to obtain a revised ticket, which justupdates Draw 1 results by showing gray movers.

After Draw 2, every remaining hanging n-Draw bet earns credit—its amountto be shown later on—. Any ticket, revised or not, containing a hangingbet can be used as bet slip to place credit bets as follows: The bettormarks to select ‘credit percentage’ 99 and either ‘new slip’ or not. Incase of no new bet slip, each existing bet-on-mover-site in Draw 3 onthe bet ticket automatically counts as a Draw 3 ‘simple’ bet, the bettormay mark to select bet-on movers in additional sites 51 to bet on. Thetotal hanging bet credit modified by selected percentage 99, referred toas r3, will be evenly applied to all Σ(#51(M)) Draw 3 ‘simple’ bets.Every original $a bet will be reduced to a $a*(100−r2)%*(100−r3)% bet.The ticket as bet slip will be approved by the computer so that arevised ticket as shown in FIG. 2G, 3G or 4G can be issued. In additionto existing data the revised ticket shows selected credit percentage forDraw 3, and all new bet-on selections marked with “=”. Besides, everybet-on mover will show up in gray to indicate its location determined byDraw 2. The bettor with no intention to make credit bets may submit ahanging bet ticket without any credit bet selection to obtain a revisedticket, which just updates Draw 2 outcomes by showing gray movers.

After Draw 3, every remaining hanging n-Draw bet earns credit—its amountto be shown later on—. Any ticket, revised or not, containing a hangingbet can be used as bet slip to place credit bets as follows: The bettormarks to select ‘credit percentage’ 99 and either ‘new slip’ or not. Incase of no new bet slip, each existing bet-on-mover-site in Draw 3 onthe bet ticket automatically counts as a Draw 4 ‘simple’ bet, the bettormay mark to select bet-on movers in additional sites 61 to bet on. Thetotal hanging bet credit modified by selected percentage 99, referred toas r4, will be evenly applied to all Σ(#61(M)) Draw 4 ‘simple’ bets.Every original $a bet will be reduced to a$a*(100−r2)%*(100−r3)%*(100−r4)% bet. The ticket as bet slip will beapproved by the computer so that a revised ticket as shown in FIG. 2H.(Note that the drawings presented here do not include FIGS. 3H and 4H toshow a possible revised 4-Draw ticket using bet slip as shown in FIGS.3AA, 4AA). In addition to existing data the revised ticket showsselected credit percentage for Draw 4, and all new bet-on selectionsmarked with “=”. Besides, every bet-on mover will show up in gray toindicate its location determined by Draw 3. The bettor with no intentionto make credit bets may submit a hanging bet ticket without any creditbet selection to obtain a revised ticket, which just updates Draw 3outcomes by showing gray movers.

Regardless of after which Draw, in the case of selecting ‘new slip’, thebettor submits the ticket without any credit bet selection to receive anon-cashable credit voucher together with a revised bet ticket. On therevised ticket, selected credit percentage will show up, and Drawoutcomes with gray movers will be printed. The voucher shows besidescredit amount an index y, which indicates a carryover inverse of theproduct of all winning probabilities in the submitted ticket. Thevoucher can be used like cash for placing bets on a new slip, on which‘credit’ must be marked. Later on in the calculation of payoff, houseedge e(x) will be a function of x=y*z, where z is the inverse of theproduct of all winning probabilities in the new ticket.

Random Number Generator

The game requires a manipulation-proof random number generator to pickruled movements. It can be a mechanical device like the one used atkeno. While there a number on each ball, here a symbol representing oneruled movement. While there one cage with 80 balls, here one cage foreach mover in which equally many, say, four or five balls for eachmovement. The generator can also be TIMER functioned using a clock with8,640,000 centi-seconds per day so that every centi-second is assignedto one movement such as it will be movement ‘X’ when 3,456,789centi-seconds have elapsed since midnight. Which centi-second isassigned to which movement can be made known to the public. There is nofear of manipulation because pressing a button mechanically by a forgernobody is able to catch a desired elapsed centi-second of a day. Anyway,the generator must obviously produce all ruled movements equallyprobable at random.

Description of the Non-Automatic Game

The game requires at least one TV/computer monitor, several wageringmachines connected to a data processing computer, printed paper betslips and random number generators. Naturally, the connection betweenwagering machines and the data processing computer can be via internet.

The game requires a player, called operator, to start by putting movers12 in sites 11 arbitrarily as shown in FIGS. 2, 3 and 4. All otherplayers, called bettors, use paper or on-screen slip as shown in FIGS.2A, 3A, 3AA, 4A and 4AA to place bets as described above. At a presettime, independent of wagering activity, the operator uses randomgenerators as described above to execute the first draw, called Draw #1,one movement for each mover. The outcomes will be displayed on themonitor as shown in FIGS. 2, 3 and 4. Besides, in case of using playingsurface as shown in FIG. 3 or 4, the display also includes FIG. 3′ or 4′respectively. In FIG. 3′ and 4′, movers locate in gray sites before Drawand become gray after Draw. The outcomes will also be input into thecomputer to determine if any bet ticket contains selections matching theoutcomes so that its holder can obtain payoff or credit.

Whether there is a draw in action or not, whether having placed betsbefore Draw #1 or not, any bettor can place bets anytime just likebefore Draw #1. Besides, it is an option that the hanging ticket holdercan place credit bets as described above. At a preset time the operatorexecutes the next draw, called Draw #2, for all movers. The outcomeswill be displayed and data proceeded just like after Draw #1. As theflowchart in FIG. 1 shows, the above steps repeat. Unless pause or stophas been regulated ahead, the process will go on indefinitely, while anybettor may start or stop betting anytime. The Draw # will growaccordingly. But bettors don't need to pay attention to it. For the sakeof reference, ‘Draw 1 is Draw # so and so’ will be printed on every betticket. A regulated stop must allow any existing bets to reach finalresults.

Description of the Automatic Version

To play the automatic version, one needs a video game machine orpersonal computer equipped with made-to-order software inclusive randomnumber generator such as the TIMER functioned one described above totake care of drawing ruled movements. Other than the non-automatic game,every player is operator as well as bettor. Each draw is effective onlyto movers of the concerning playing surface. There will be no paper betslip or ticket. But certainly as an option a printer can be connected toprint out anything displayed on monitor. The hardware includes apointing device or touch screen monitor for the player to make/markselections.

The game starts with the display of a playing surface as shown in FIG.2, 3 or 4 with additional icons/items named “Another playing surface”,“Bet slip” and “Account”.

Clicking any item on the display screen will either highlight it orresult in a new display.

Clicking a highlighted item is to cancel that selection.

Clicking “Another playing surface” will result in the display of anotherone. All playing surfaces as shown in FIGS. 2 to 4, or maybe some onenot given here, will be displayed cyclically one after another ifclicking “Another playing surface” continues.

Clicking “Bet slip” will display a bet slip as shown in FIG. 2A, 3A,3AA, 4A or 4AA with additional icons “Ticket” and “Account”; andfurthermore “Alternative slip” in case of playing surface being as shownin FIG. 3 or 4.

Clicking “Alternative slip” will switch to a chain bet slip if thedisplayed one is for 1-Draw bet, or conversely. That is, switchingbetween 3A and 3AA or 4A and 4AA.

The player places bets on screen just as on paper in the non-automaticgame; then clicks “Ticket” to submit. If the submitted slip isincomplete or contains error, there will be a message such as‘Incomplete! Please select one per bet amount or total bet amount’,requiring the bettor to make amendment. If the submission is approved, abet ticket with a ticket number as shown in FIGS. 2B to 2E, 3B to 3E or4B to 4E, with additional icons “Go back”, “Cancel”, “Draw” and“Account” will show up.

Facing a bet ticket:

Clicking “Go back” allows the player to return the submitted slip tomake changes.

Clicking “Cancel” is to abandon the submitted slip and to request ablank bet slip.

Clicking “Account’ will result in a display as shown in FIG. 5. It showsavailable balance, credit voucher data, all bet tickets with status andicons “Playing surface’, “Bet slip” and “Exit”.

Clicking “Draw” will cause a draw and computer data processing.Subsequently, the Draw 1 outcomes will automatically update theconcerning bet ticket by painting bet-on movers in gray. Note that otherthan in non-automatic game, non-bet-on movers will neither show up norbe involved in the draw. After update with gray mover, “Go back” and“Cancel” disappear. There are three possibilities:

(1) In case of 1-Draw bet ticket or n-Draw one with no hanging bet,there will be an icon “Account”.

(2) In case of 1-Draw bet ticket containing more than Draw 1, there willbe icons “Draw” and “Account”. The player can click “Draw” to executeanother draw so that, whatever applicable, Draw 2 to 4 outcomes willautomatically update the concerning bet ticket by painting bet-on moversin gray. Each time, “Account” remains available.

(3) In case of n-Draw bet ticket with hanging bets, there will be icons“Submit”, “Draw” and “Account”. The player can mark to make credit betselections just as in the non-automatic game, and then click “Submit” toreceive a revised ticket as in the non-automatic game as shown in FIG.2F, 3F or 4F. The display of it will include icon “Draw” and “Account”.

In case of using ‘new slip’, the player will see a message requiring toclick “Account” and to click one credit voucher as fund for ‘new slip’bets.

Facing an “Account” display as shown in FIG. 5:

Clicking a credit voucher will return to new slip display.

Clicking a certain ticket will return to the display of that ticket forviewing it or placing credit bets or executing a “Draw”.

The display of any finished ticket includes icon “Account”. The displayof any unfinished 1-Draw ticket includes icons “Draw” and “Account”. Thedisplay of any hanging n-Draw ticket includes icons “Submit”, “Draw” and“Account”.

The player continues just as in the non-automatic game, except thatclicking “Draw” is required to execute a Draw. After a Draw, outcomeswill automatically update the concerning bet ticket by painting bet-onmovers in gray.

Clicking “Submit” on hanging bet ticket after selecting none or severalcredit bets is to submit it for approval. After approval, the icon“Submit” disappears so that the player can click “Draw” or “Account” tocontinue.

Facing an “Account” display and clicking “Playing surface” or “Bet slip”allows the bettor to continue in whichever way preferred, while “Exit”to end the game.

Calculation of Probabilities

There are 4 categories of probability formulae according to bet typesand playing surfaces.

(1) 1-Draw ‘jump’ bets using slip as shown in FIG. 2A. Here we have w=7.

Although probabilities in this category are actually independent ofwhich Draw, for practical purpose we need n=1 to 4 to specify Draw. AnyDraw n ‘simple’ bet has winning probability pn=1/w. Any Draw n site’ bethas winning probability pn=1/ŵm, where m is the number of bet-on moversin that selected site of the concerning draw. Any Draw n ‘mixed’ bet haswinning probability pn=1/ŵm where m is the number of bet-on movers ofthe concerning draw.

(2) 1-Draw ‘cylinder’ bets using slip as shown in FIG. 3A. Here we havew=6.

Referring to FIG. 3, the identification of top border line 1001 withbottom border line 1002 allows us to assign any one of the ten siteswith 1-dimensional coordinates x and others with coordinates x+i, whereevery calculation involving i or x is modulo 10 arithmetic. Now we canreplace A, B, etc by x, x+1, etc. respectively. —For example, A:0, B:9,C:8, D:7, E:6, F:5, G:4, H:3, K:2, L:1.— And we can also say that x+ilies i sites away from x. A movement from 0 to i is equivalent to amovement from x to x+i. Thus,

-   -   ‘00’ moves a mover from x to x, defining a 1-movement path        d1(0),    -   ‘U1’ moves a mover from x to x+1, defining a 1-movement path        d1(1),    -   ‘U2’ moves a mover from x to x+2, defining a 1-movement path        d1(2),    -   ‘U3’ moves a mover from x to x+3, defining a 1-movement path        d1(3),    -   ‘D1’ moves a mover from x to x+9, defining a 1-movement path        d1(9),    -   ‘D2’ moves a mover from x to x+8, defining a 1-movement path        d1(8).

There is no other 1-movement path d1(i).

Let #d1(i) denote the number of all d1(i) for i. Obviously, there are

-   -   #d1(0)=#d1(1)=#d1(2)=#d1(3)=#d1(8)=#d1(9)=1 and        #d1(4)=#d1(5)=#d1(6)=#d1(7)=0; in total 6.

p1M=#d1(i(M))/w is the probability of mover M from its start location toget on a d1(i(M)) path to reach the site lying i(M) sites away. Here weneed i(M) to specify i for the concerning M, though all i(M) areidentical regardless of which M.

Let d2(i) be any d1(x) followed by any d1(i−x), defining a 2-movementpath from any site to a site lying i sites away.

Let #d2(i) denote the number of all d2(i) for i. #d2(i) is the sum of#d1(x)*#d1(i−x) over all x. Explicitly, there are

-   -   #d2(0)=5, #d2(1)=6, #d2(2)=5, #d2(3)=4, #d2(4)=3, #d2(5)=2,        #d2(6)=2, #d2(7)=2, #d2(8)=3, #d2(9)=4; in total 36, that is 6̂2.

p2M=#d2(i(M))/ŵ2 is the probability of mover M from its start locationto get on a d2(i(M)) path to reach the site lying i(M) sites away. Herewe need i(M) to specify i for the concerning M, though all i(M) areidentical regardless of which M.

Let d3(i) be any d1(x) followed by any d2(i−x), defining a 3-movementpath from any site to a site lying i sites away.

Let #d3(i) denote the number of all d3(i) for i. #d3(i) is the sum of#d1(x)*#d2(i−x) over all x. Explicitly, there are

-   -   #d3(0)=25, #d3(1)=27, #d3(2)=27, #d3(3)=25, #d3(4)=22,        #d3(5)=18, #d3(6)=16, #d3(7)=16, #d3(8)=18, #d2(9)=22; in total        216, that is 6̂3.

p3M=#d3(i(M))/ŵ3 is the probability of mover M from its start locationto get on a d3(i(M)) path to reach the site lying i(M) sites away. Herewe need i(M) to specify i for the concerning M, though all i(M) areidentical regardless of which M.

Let d4(i) be any d1(x) followed by any d3(i−x), defining a 4-movementpath from any site to a site lying i sites away.

Let #d4(i) denote the number of all d4(i) for i. #d4(i) is the sum of#d1(x)*#d3(i−x) over all x. Explicitly, there are

-   -   #d4(0)=135, #d4(1)=144, #d4(2)=148, #d4(3)=144, #d4(4)=135,        #d4(5)=124, #d4(6)=115, #d4(7)=112, #d4(8)=115, #d4(9)=124; in        total 1296, that is 6̂4.

p4M=#d4(i(M))/ŵ4 is the probability of mover M from its start locationto get on a d4(i(M)) path to reach the site lying i(M) sites away. Herewe need i(M) to specify i for the concerning M, though all i(M) areidentical regardless of which M.

A Draw n ‘simple’ bet, where n=1 to 4, on mover M lying i(M) sites awaywill be denoted by dn(i(M)). It has winning probabilitypn=pnM=#dn(i(M))/ŵn.

A Draw n ‘site’ bet, where n=1 to 4, on movers M lying each i(M) sitesaway from site S will be denoted by dnS( . . . , i(?), . . . ), where ?goes from mover #1 to #6, and i(?) is i(M) if ? is a bet-on mover,otherwise ‘-’ (a dash). For example, d3B(-, 2, 1, -, -, 8) is a Draw 3‘site’ bet on site B with bet-on movers #2, #3 and #6, lyingrespectively 2, 1 and 8 sites away from site B. Or, d4E(3, 2, 1, -, 7,-) is a Draw 4 ‘site’ bet on site E with bet-on movers #1, #2, #3 and#5, lying respectively 3, 2, 1 and 7 sites away from site E. The dnS( .. . , i(?), . . . ) bet has the winning probability of pn=Π(pnM), wheremultiplication is over all bet-on movers M in site S of Draw n.

A Draw n ‘mixed’ bet, where n=1 to 4, on movers M lying each i(M) sitesaway will be denoted by dnX( . . . , i(?), . . . ), where ? goes frommover #1 to #6, and i(?) is i(M) if ? is a bet-on mover, otherwise ‘-’(a dash). For example, d2X(2, 5, -, -, -, 6) is a Draw 2 ‘mixed’ betwith bet-on movers #1, #2 and #6, lying respectively 2, 5 and 6 sitesaway. Or, d3X(-, -, 3, 2, 3, -) is a Draw 3 ‘mixed’ bet with bet-onmovers #3, #4 and #5, lying respectively 3, 2 and 3 sites away. Or,d4X(-, (3,4), (2,1), -, -, (00)) is a Draw 4 ‘mixed’ bet with bet-onmovers #2, #3 and #6, lying respectively (3,4), (2,1) and (00) sitesaway. The dnS( . . . , i(?), . . . ) bet has the winning probability ofpn=Π(pnM) where multiplication is over all bet-on movers of Draw n.

(3) 1-Draw ‘torus’ bets using slip as shown in FIG. 4A. Here we havew=9.

Referring now to FIG. 4, the identification of top border line 1001 withbottom border line 1002 and left border line 1003 with right border line1004 allows us to assign any one of the twenty sites with matrixcoordinates (x,y) and others with coordinates (x+i,y+j), where everycalculation involving i or x is modulo 4 arithmetic, involving j or y ismodulo 5 arithmetic. Now we can replace AA, AB, etc. by (x,y), (x,y+1)etc. respectively. —For example, AA:(0,0), AB:(0,1), AC:(0,2), AD:(0,3),AE:(0,4), BA:(1,0), BB:(1,1), BC:(1,2), BD:(1,3), BE:(1,4), CA:(2,0),CB:(2,1), CC:(2,2), CD:(2,3), CE:(2,4), DA:(3,0), DB:(3,1), DC:(3,2),DD:(3,3), DE:(3,4).— And we can also say that (x+i,y+j) lies (i,j) sitesaway from (x,y). A movement from (0,0) to (i,j) is equivalent to amovement from (x,y) to (x+i,y+j). Thus,

-   -   ‘00’ moves a mover from (x,y) to (x,y), defining a 1-movement        path d1(0,0).    -   ‘E’ moves a mover from (x,y) to (x,y+1), defining a 1-movement        path d1(0,1)    -   ‘W’ moves a mover from (x,y) to (x,y+4), defining a 1-movement        path d1(0,4)    -   ‘N’ moves a mover from (x,y) to (x+3,y), defining a 1-movement        path d1(3,0).    -   ‘S’ moves a mover from (x,y) to (x+1,y), defining a 1-movement        path d1(1,0).    -   ‘NE’ moves a mover from (x,y) to (x+3,y+1), defining a        1-movement path d1(3,1).    -   ‘SE’ moves a mover from (x,y) to (x+1,y+1), defining a        1-movement path d1(1,1).    -   ‘NW’ moves a mover from (x,y) to (x+3,y+4), defining a        1-movement path d1(3,4).    -   ‘SW’ moves a mover from (x,y) to (x+1,y+4), defining a        1-movement path d1(1,4).

There is no other 1-movement path d1(i,j).

Let #d1(i,j) denote the number of all d1(i,j) paths from (0,0) to (i,j).Obviously, there are

-   -   #d1(0,0)=#d1(0,1)=#d1(0,4)=#d1(1,0)=#d1(1,1)=#d1(1,4)=#d1(3,0)=#d1(3,1)=#d1(3,4)=1        and #d1(i,j)=0 for all other (i,j); in total 9.

p1M=#d1(i(M),j(M))/w is the probability of mover M from its startlocation to get on a d1(i(M),j(M)) path to reach the site lying(i(M),j(M)) sites away. Here we need (i(M),j(M)) to specify (i,j) forthe concerning M, though all (i(M),j(M)) are identical regardless ofwhich M.

Let d2(i,j) be any d1(x,y) followed by any d1(i−x,j−y), defining a2-movement path from any site to a site lying (i,j) sites away.

Let #d2(i,j) denote the number of all d2(i,j) paths from (0,0) to (i,j).#d2(i,j) is the sum of #d1(x,y)*#d1(i−x,j−y) over all x and y.Explicitly, there are

-   -   #d2(0,0)=9, #d2(0,1)=6, #d2(0,2)=3, #d2(0,3)=3, #d2(0,4)=6,        #d2(1,0)=6, #d2(1,1)=4, #d2(1,2)=2, #d2(1,3)=2, #d2(1,4)=4,        #d2(2,0)=6, #d2(2,1)=4, #d2(2,2)=2, #d2(2,3)=2, #d2(2,4)=4,        #d2(3,0)=6, #d2(3,1)=4, #d2(3,2)=2, #d2(3,3)=2, #d2(3,4)=4; in        total 81, that is 9̂2.

p2M=#d2(i(M),j(M))/ŵ2 is the probability of mover M from its startlocation to get on a d2(i(M),j(M)) path to reach the site lying(i(M),j(M)) sites away. Here we need (i(M),j(M)) to specify (i,j) forthe concerning M, though all (i(M),j(M)) are identical regardless ofwhich M.

Let d3(i,j) be any d1(x,y) followed by any d2(i−x,j−y), defining a3-movement path from any site to a site lying (i,j) sites away.

Let #d3(i,j) denote the number of all d3(i,j) paths from (0,0) to (i,j).#d3(i,j) is the sum of #d1(x,y)*#d2(i−x,j−y) over all x and y.Explicitly, there are

-   -   #d3(0,0)=49, #d3(0,1)=42, #d3(0,2)=28, #d3(0,3)=28, #d3(0,4)=42,        #d3(1,0)=49, #d3(1,1)=42, #d3(1,2)=28, #d3(1,3)=28, #d3(1,4)=42,        #d3(2,0)=42, #d3(2,1)=36, #d3(2,2)=24, #d3(2,3)=24, #d3(2,4)=36,        #d3(3,0)=49, #d3(3,1)=42, #d3(3,2)=28, #d3(3,3)=28, #d3(3,4)=42,        in total 729, that is 9̂3.

p3M=#d3(i(M),j(M))/ŵ3 is the probability of mover M from its startlocation to get on a d3(i(M),j(M)) path to reach the site lying(i(M),j(M)) sites away. Here we need (i(M),j(M)) to specify (i,j) forthe concerning M, though all (i(M),j(M)) are identical regardless ofwhich M.

Let d4(i,j) be any d1(x,y) followed by any d3(i−x,j−y), defining a4-movement path from any site to a site lying (i,j) sites away.

Let #d4(i,j) denote the number of all d4(i,j) paths from (0,0) to (i,j).#d4(i,j) is the sum of #d1(x,y)*#d3(i−x,j−y) over all x and y.Explicitly, there are

-   -   #d4(0,0)=399, #d4(0,1)=357, #d4(0,2)=294, #d4(0,3)=294,        #d4(0,4)=357, #d4(1,0)=380, #d4(1,1)=340, #d4(1,2)=280,        #d4(1,3)=280, #d4(1,4)=340, #d4(2,0)=380, #d4(2,1)=340,        #d4(2,2)=280, #d4(2,3)=280, #d4(2,4)=340, #d4(3,0)=380,        #d4(3,1)=340, #d4(3,2)=280, #d4(3,3)=280, #d4(3,4)=340; in total        6561, that is 9̂4.

p4M=d4(i(M),j(M))/ŵ4 is the probability of mover M from its startlocation to get on a d4(i(M),j(M)) path to reach the site lying(i(M),j(M)) sites away. Here we need (i(M),j(M)) to specify (i,j) forthe concerning M, though all (i(M),j(M)) are identical regardless ofwhich M.

A Draw n ‘simple’ bet, where n=1 to 4, on mover M lying each (i(M),j(M))sites away will be denoted by dn((i(M),j(M)). It has winning probabilitypn=pnM.

A Draw n ‘site’ bet, where n=1 to 4, on movers M lying each (i(M),j(M))sites away from site S will be denoted by dnS( . . . , (i(?),j(?)), . .. ), where ? goes for each (i,j) from movers #1 to #6, and (i(?),j(?))is (i(M),j(M)) if ? is a bet-on mover, otherwise ‘-’ (a dash). Forexample, d3BB(-, (2,3), (3,1), -, -, (0,4)) is a Draw 3 ‘site’ bet onsite BB with bet-on movers #2, #3 and #6, lying respectively (2,3),(3,1) and (0,4) sites away from site BB. Or, d4DA((3,3), (0,2), (1,1),-, (0,0), -) is a Draw 4 ‘site’ bet on site DA with bet-on movers #1,#2, #3 and #5, lying respectively (3,3), (0,2), (1,1) and (0,0) sitesaway from site DA. The dnS( . . . , (i(?),j(7)), . . . ) bet has winningprobability pn=Π(pnM), where multiplication is over all bet-on movers Min site S of Draw n.

A Draw n ‘mixed’ bet, where n=1 to 4, on movers M lying each (i(M),j(M))sites away will be denoted by dnX( . . . , (i(?),j(?)), . . . ), where ?goes for each (i,j) from mover #1 to #6, and (i(?),j(?)) is (i(M),j(M))if ? is a bet-on mover, otherwise ‘-’ (a dash). For example, d2X((3,2),(2,3), -, -, -, (1,1)) is a Draw 2 ‘mixed’ bet with bet-on movers #1, #2and #6, lying respectively (3,2), (2,3) and (1,1) sites away. Or, d3X(-,-, (0,3), (1,2), (3,4), -) is a Draw 3 ‘mixed’ bet with bet-on movers#3, #4 and #5, lying respectively (0,3), (1,2) and (3,4) sites away. ThednX( . . . , (i(?),j(?)), . . . ) bet has winning probability pn=Π(pnM),where multiplication is over all bet-on movers M of Draw n.

(4) n-Draw ‘chain’ bets for n=2 to 4. Here we have w=7, 6 or 9 in caseof using bet slip as shown in FIG. 2A, 3AA or 4AA respectively.

A 2-Draw bet with m bet-on movers has probability p=1/ŵm to becomehanging, and probability p=1/ŵ2m to win.

A 3-Draw bet with m bet-on movers has probability p=1/wAm to becomehanging, and probability p=1/ŵ2m to remain hanging, and probabilityp=1/ŵ3m to win.

A 4-Draw bet with m bet-on movers has probability p=1/ŵm to becomehanging, and probability p=1/ŵ2m to remain hanging, and probabilityp=1/ŵ3m to remain hanging once more, and probability p=1/ŵ4m to win.

Besides, a ticket has probabilities:

p1=Π(#31(M))/ŵm to be hanging after Draw 1, then

p2=Π(#41(M))/ŵm to win or remain hanging after Draw 2, then

p3=Π(#51(M))/ŵm and p4=Π(#61(M))/ŵm to win or remain hanging after Draw3, then

p4=Π(#61(M))/ŵm to win at Draw 4.

House Edges, Payoffs and Credits

This game requires reasonable house edges such as follows: Let x be theinverse of the product of winning probabilities of all involved draws.

e(x)=2.5+x/4 for 1≦x≦10

e(x)=4+(n+1)[n/2+(x−10̂n)/90(10̂n)] for 10<x with integer n satisfying10̂n<x≦10̂(n+1).

For 1-Draw bets with winning probabilities p1 to p4 of Draw 1 to 4respectively as declared in ‘Calculation of probabilities’ according toplaying surfaces being used:

A winning $a Draw 1 1-Draw bet pays $a*(100−e(1/p1))%/p1.

A winning $a Draw 2 1-Draw bet pays $a*(100−e(1/p2))%/p2.

A winning $a Draw 3 1-Draw bet pays $a*(100−e(1/p3))%/p3.

A winning $a Draw 4 1-Draw bet pays $a*(100−e(1/p4))%/p4.

For multi-draw bets with m bet-on movers let r2, r3 and r4 be thepercentage points of credit selected by the bettor for Draw 2, 3 and 4respectively.

A winning $a 2-Draw bet pays $a*(100−r2)%*(100−e(ŵ2m))%/ŵ2m.

A winning $a 3-Draw bet pays $a*(100−r2)%(100−r3)%*(100−e(ŵ3m))%/ŵ3m.

A winning $a 4-Draw bet pays$a*(100−r2)%*(100−r3)%*(100−r4)%*(100−e(ŵ4m))%/ŵ4m

In the case of calculating credits and credit bet payoffs, we use thetotal amount of a ticket rather than per bet amount. Let $aa be thetotal bet amount on a ticket.

After Draw 1, a hanging $aa ticket earns total credit $aa/p1. Eachwinning Draw 2 credit ‘simple’ bet pays$aa*r2%*w*(e(ŵ(m+1)))/(p1*Σ(#41(M))).

After Draw 2, a hanging $aa ticket earns total credit$aa*(100−r2)%/(p1*p2). Each winning Draw 3 credit ‘simple’ bet pays$aa*(100−r2)%*r3%*w*(e(ŵ(2m+1)))/(p1*p2*Σ(#51(M))).

After Draw 3, a hanging $aa ticket earns total credit$aa*(100−r2)%*(100−r3)%/(p1*p2*p3). Each winning Draw 4 credit ‘simple’bet pays$aa*(100−r2)%*(100−r3)%*r4%*w*(e(ŵ(3m+1)))/(p1*p2*p3*Σ(#61(M))).

NUMERICAL EXAMPLES

In order to make calculations less complex, no house edge will beapplied below.

In the ‘simple’ bet ticket as shown in FIG. 2B, in any Draw n, every bethas probability pn=1/7 to win payoff $2/pn=$14.

In the ‘simple’ ticket as shown in FIG. 3B, in Draw 1, every bet hasprobability p1=1/6, to win $2/p1=$12; in Draw n, where n=2 to 4, everydn(i(M)) bet has probability pn=#dn(i(M))/ŵn to win payoff $2/pn. Forexample, d2(8(3)) bet has p2=3/6̂2; d3(4(4)) bet has p3=22/6̂3; d4(2(2))bet has p4=148/6̂4.

In the ‘simple’ ticket as shown in FIG. 4B in Draw 1, every bet hasprobability p1=1/9 to win $1/p1=$9; in Draw n, where n=2 to 4, everydn(i(M), j(M)) bet has probability pn=#dn(i(M),j(M))/ŵn to win payoff$1/pn. For example, d2(0(2),1(2)) bet has p2=6/9̂2; d3(2(5),1(5)) bet hasp3=36/9̂3; d4(3(6),3(6)) bet has p4=280/9̂4.

In the ‘site’ ticket as shown in FIG. 2C, in any Draw n, every bet hasprobability pn=1/7̂m, where m is the number of bet-on movers in theselected site, to win payoff $2/pn. For example, in Draw 1, Site D′ bethas probability p1=1/7̂3, in Draw 2, Site ‘C’ bet has probabilityp2=1/7̂2, in Draw 3, Site ‘F’ bet has probability p3=1/7̂2, in Draw 4,Site ‘B’ bet has probability p4=1/7̂3.

In the ‘site’ ticket as shown in FIG. 3C, every dnS( . . . , i(M), . . .) bet, where n=1 to 4, has probability pn=Π(pnM), where multiplicationis over all bet-on movers M of Draw n, to win payoff $2/pn. For example,d1C(1, -, -, -, 0, -) bet has p1=#d1(1)*#d1(0)/6̂2=1/6̂2; d2H(6, 0, 8, 3,0, 0) bet has p2=#d2(6)*#d2(8)*#d2(3)/6̂(2*3)=2*3*4/6̂6; d3G(7, 1, 9, 4,-, -) bet has p3i=#d3(7)*#d3(1)*#d3(9)*#d3(4)/6̂(3*4)=16*27*22*25/6̂12;d4F(8, -, 0, 5, 7, 0) bet hasp4=#d4(8)*#d4(0)*#d4(5)*#d4(7)*#d4(0)/6̂(4*5)=115*135*112*124*135/6̂20.

In the ‘site’ ticket as shown in FIG. 4C, every dnS( . . . ,(i(M),j(M)), . . . ) bet, where n=1 to 4, has probability pn=Π(pnM),where multiplication is over all bet-on movers M in site S, to winpayoff $2/pn. For example, d1BB((3,4), -, -, (3,1), (1,0), -) bet hasp1=#d1(3,4)*#d1(3,1)*#d1(1,0)/9̂3=1/9̂3; d2CB(-, -, (1,2), -, (2,0), -)bet has p2=#d2(1,2)*#d2(2,0)/9̂(2*2)=2*6/9̂4; d3BA((1,2), -, -, -, -,(2,2)) bet has p3=#d3(1,2)*#d3(2,2)/9̂(3*2)=28*24/9̂6; d4DE((3,1), -,(2,0), -, -, (0,1)) bet hasp4=#d4(3,1)*#d4(2,0)*#d4(0,1)/9̂(4*3)340*380*357/9̂12.

In the ‘mixed’ ticket as shown in FIG. 2D, every bet, in Draw 1, hasprobability p1=1//7̂4 to win payoff $100/(p1*Π(p1M)), whereΠ(p1M)=6*5*5*6=900; in Draw 2 probability p2=1/7̂5 to win payoff$100/(p2*Π(p2M)), where Π(p2M)=5*5*6*6*5=3600; in Draw 3 probabilityp3=1/7̂5 to win payoff $100/(p3*Π(p3M)), where Π(p3M)=5*6*5*4*6=3000; inDraw 4 probability p4=1/7̂4 to win payoff $100/(p1*Π(p4M)), whereΠ(p4M)=6*4*4*6=576.

In the ‘mixed’ ticket as shown in FIG. 3D, every dnX( . . . , i(M), . .. ) bet, where n=1 to 4, has probability pn=Π(pnM), where multiplicationis over all bet-on movers M of the concerning draw, to win payoff $0,10/pn. For example, d1X(0, -, 1, 3, -, 9) bet hasp1=#d1(0)*#d1(1)*#d1(3)*#d1(9)/6̂4=(1*1*1*1)/6̂4; d2X(1, 4, -, 8, 7, 6)bet has p2=#d2(1)*#d2(4)*#d2(8)*#d2(7)*#d2(2)[/6̂(2*5)=6*3*3*2*2/6̂10;d3X(-, 3, 8, -, 5, 1) bet hasp3=#d3(3)*#d3(8)*#d3(5)*#d3(1)/6̂(3*4)=25*18*18*27/6̂12; d4X(-, 1, 9, 3,0, 2) bet hasp4=#d4(1)*#d4(9)*#d4(3)*#d4(0)*#d4(2)/6̂(4*5)=144*124*144*135*148/6̂2

In the ‘mixed’ ticket as shown in FIG. 4D, every dnX( . . . ,(i(M),j(M)), . . . ) bet, where n=1 to 4, has probability pn=Π(pnM),where multiplication is over all bet-on movers of that draw, to winpayoff $0, 10/pn. For example, d1X((3,4), -, (3,0), (1,0), (3,4), -) bethas p1=#d1(3,4)*#d1(3,0)*#d1(1,0)*#d1(3,4)/9̂4=1*1*1*1/9̂4; d2X(-, (3,0),-, -, (2,1), -) bet has p2=#d2(3,0)*#d2(2,1)/9̂(2*2)=6*4/9̂4; d3X((2,1),-, -, -, (0,0), (0,3) bet hasp3=#d3(2,1)*#d3(0,0)*#d3(0,3)/9̂(3*3)=26*49*28/9̂9; d4X((1,4), (3,1), -,(0,3), -, (3,3)) bet hasp4=#d4(1,4)*#d4(3,1)*#d4(0,3)*#d4(3,3)/9̂(4*4)340*340*294*280/9̂16.

The 4-Draw ticket as shown in FIG. 2E—here w=7—has probabilityp1=4*4*5*5/7̂4=400/2401 to become hanging and earn Draw 2 credit$200/p1=$1,200.50; then probability p2=3*2*4*4/7̂4=96/2401 to remainhanging, and earn Draw 3 credit $200*(100−r2)%/(p1*p2); then probabilityp3=3*4*3*3/ŵ4=108/2401 to remain hanging, and earn Draw 4 credit$200*(100−r2)%*(100−r3)%/(p1*p2*p3); and finally probabilityp4=2*2*2*2/ŵ4=16/2401 to win payoff$200*(100−r2)%*(100−r3)%*(100−r4)%/(p1*p2*p3*p4); which is$100,166,770.86 if r2=r3=r4=0. All p1 to p4 hold in revised tickets.

The revised 4-Draw ticket as shown in FIG. 2F with Σ(#41(M))=23 credit70% bets contains four credit bet winners, each paying$200*70%*7/(p1*23)=$255.76. It also has probability p2 to earn Draw 3credit $200*(100−70)%/(p1*p2)=$9,007.50; and then probability p3 toremain hanging and earn Draw 4 credit$200*(100−70)%*(100−r3)%/(p1*p2*p3).

The revised 4-Draw ticket as shown in FIG. 2G with Σ(#51(M)))=25 credit60% bets contains four credit bet winners, each paying$200*70%*60%*7/(p1*p2*25)=$1513.26. It also has probability p3 to earnDraw 4 credit $200*(100−70) %*(100−60)%/(p1*p2*p3)=$80,100.04.

The revised 4-Draw ticket as shown in FIG. 2H with Σ(#61(M)=28 credit80% bets will result in four credit bet winners, each paying$200*(100−70)%*(100−60)%*80%*7/(p1*p2*p3*28)=$16,200.16. It also hasprobability p4 to win payoff$200*(100−70)%*(160−60)%*(100−80)%/(p1*p2*p3*p4)=$2,404,002.60 on oneoriginal bet.

The 3-Draw ticket as shown in FIG. 3E—here w=6—has probabilityp1=5*5*5*5*5/6̂5=35/7776 to become hanging and earn Draw 2 credit$200/p1=$497.66; then probability p2=4*4*4*4*4/6̂5=1024/7776 to remainhanging, and earn Draw 3 credit $200*(100−r2)%/(p1*p2); and finallyprobability p3=3*2*3*2*3/6̂5=108/7776 to win payoff$200*(100−r2)%*(100−r3)%/(p1*p2*p3); which is $272,097.79 if r2=r3=0.All p1 to p3 hold in revised tickets.

The revised 3-Draw ticket as shown in FIG. 3F with Σ(#41(M))=30 credit70% bets assures five credit bet winners, each paying$200*70%*6/(p1*30)=$69.67. It also has probability p2 to earn Draw 3credit $200*(100−70)%/(p1*p2)=$113.37.

The revised 3-Draw ticket as shown in FIG. 3G with Σ(#51(M))=30 credit80% bets assures five credit bet winners, each paying$200*(100−70)%*80%*6/(p1*p2*30)=$11.32. It also has probability p3 towin payoff $200*%*(100−70)*%*(100−80)/(p1*p2*p3)=$16,325.87 on oneoriginal bet.

The 3-Draw ticket as shown in FIG. 4E—here w=9—has probabilityp1=8*8*8*7/9̂4=4704/6561 to become hanging, and earn Draw 2 credit$200/p1=$278.95; then probability p2=7*7*6*6/9̂4=1764/6561 to remainhanging, and earn Draw 3 credit $200*(100−r2)%/(p1*p2); and finallyprobability p3=6*5*5*5/9̂4=750/6561 to win payoff$200*(100−r2)%*(100−r3)%/(p1*p2*p3); which is $9,076.39 if r2=r3=0. Allp1 to p3 hold in revised tickets.

The revised 3-Draw ticket as shown in FIG. 4F with Σ(#41(M))=35 credit60% bets contains four credit bet winners, each paying$200*60%*9(p1*35)=$43.04. It also has probability p2 to earn Draw 3credit $200*(100−60)%/(p1*p2)=$415.02.

The revised 3-Draw ticket as shown in FIG. 4G with Σ(#51(M))=34 credit70% bets will result in two to four credit bet winners, each paying$200*(100−60)%*70%*9/(p1*p2*34)=$76.90. It also has probability p3 towin payoff $200*%*(100−60)*%*(100−70)/(p1*p2*p3)=$1,089.17 on oneoriginal bet.

Now a simple example of using house edges e(1/p):

Let the playing surface be FIG. 2 and budget $80. I'll play no more thantwo Draws according to plan (1) plus (2) depending on weather and where#1 and #2 go in Draw 1.

(1) If it doesn't rain, then I place 4 $20 1-Draw ‘mixed’ bets on movers#1 and #2 moving to sites A or B in Draw1. In this case, p=(1/7)̂2, n−1,e(1/p)=5.87, there are 4 chances out of 49 to earn payoff$20*(1/p)*(100−e)%=$922.47.

In case of winning and good weather, I go somewhere to spend the profit.

In case of winning and starting to rain, I continue as follows; (1a) or(1b):

(1a) If Draw1 put both #1 and #2 to the same site, then (1a1) plus(1a2):

(1a1) I use 20% of $922.47 to place 4 ‘mixed’ bets on movers #1 and #2moving to sites A or B. In this case, p=(1/7)̂2, n=1, e(1/p)=5.87, thereare 4 chances out of 49 to earn payoff$(922.47/4)*20%(1/p)*(100−e)%=$2260.06*0.9413=$2124.33.

(1a2) I use 80% of $922.47 to place ‘simple’ bets on #1 and #2 moving toany site. In this case, p=1/7, n=0, e(1/p)=4.25, there are always twopayoffs of $(922.47/14)*80%*(1/p)*(100−e)%=$353.30.

(1b) If Draw1 put #1 and #2 to different sites, then (1b1) plus (1b2):

(1b1) I use 40% of $922.47 to place 4 ‘mixed’ bets on movers #1 and #2moving to sites A or B. In this case, p=(1/7)̂2, n=1, e(1/p)=5.87, thereare 4 chances out of 49 to earn payoff$(922.47/4)*40%*(1/p)*(100−e)%=$4248.66.

(1b2) I use 60% of $922.47 to place ‘simple’ bets on #1 and #2 moving toany site. In this case, p=1/7, n=0, e(1/p)=4.25, there are always twopayoffs of $(922.47/14)*60%*(1/p)*(100−e)%=$264.98.

(2) If it rains, then I place 16 $5 2-Draw ‘chain’ bets on movers #1 and#2 moving to sites A or B in each Draw. Here, p1=(1/7)̂2, there are 4chances out of 49 to be hanging, each with total credit4*$5*(1/p1)=$980.00.

In case of hanging, I continue as follows, (2a) or (2b), to ensure awinner:

(2a) If Draw1 put both #1 and #2 to the same site, then (2a1) plus(2a2):

(2a1) I use 80% of credit $980.00 to place Draw 2 ‘simple’ bets on #1and #2 moving to any site. In this case, p2=1/7, n=2, e(1/p1*p2)=7.81,there are always two payoffs of $(980.00/14)*80%(1/p2)*(100−e)%=$361.38.

(2a2) I keep 20% of credit $980.00 for the original bets. In this case,p2=(1/7)̂2, n=3, e(1/p1*p2)=10.62, there are 4 chances out of 49 to earnpayoff $(980.00/4)*20%*(1/p2)*(100−e)%=$2146.01.

(2b) If Draw1 put #1 and #2 to different sites, then (2b1) plus (2b2):

(2b1) I use 60% of credit $980.00 to place Draw 2 ‘simple’ bets on #1and #2 moving to any site. In this case, p2=1/7, n=2, e(1/p1*p2)=7.81,there are always two payoffs of$(980.00/14)*60%*(1/p2)*(100−e)%=$271.04.

(2b2) I keep 40% of credit $980.00 for the original bets. In this case,p2=(1/7)̂2, n=3, e(1/p1*p2)=10.62, there are 4 chances out of 49 to earnpayoff $(980.00/4)*80%*(1/p2)*(100−e)%=$4292.02.

Although (1) and (2) have everywhere the tame winning probabilities,there are different gains due to house edge being applied twice or once.House edge applied only to payoff encourages bettors to place multi-draw‘chain’ bets. By a simple scientific plan as shown in (2a1) and (2b1), ahanging bet holder can always become a sure winner.

CONCLUSION

The invention described above provides an extremely low operation costgame to be easily run by an existing or future keno/lottery kind ofoperator.

The game of invention is basically distinct from today's casinoslot/video games due to the fact that all possible outcomes with theircorresponding probabilities are made known to the public and that ituses obviously manipulation-proof random number generators. However, theautomatic version can be integrated into an existing video game machinewhere a TIMER function random number generator will be installed toreplace so-called RNG software protecting casino's profit.

The derivation of some probabilities requires modular arithmetic toproduce #d1(i) to #d4(i) and #d1(i,j) to #d4(i,j). Their values areexplicitly provided. Thus, neither the operator nor any player needs tounderstand or do the arithmetic. The computer will just apply thosevalues. Besides, there are examples to help everyone get acquaintancewith practical calculations.

To make the bettor no regret, every hanging bet earns non-cashablecredit equal to the payoff value without house edge. The holder can useit scientifically to result in a winner. The operator can make houseedge effective on the final payoff. Charging house edge only on finalpayoff makes purchasing a multi draw ticket more incentive than ticketsdraw by draw. Besides, house edge should be on a whole ticket instead ofeach single bet, and based on the ratio of payoff to the total betamount to allow lower ratio tickets enjoy lower house edges. Naturally,setting house edges is not inventor's business, but the game'spopularity depends on reasonable house edges such as provided above.Knowing all possible outcomes with corresponding probabilities and fairhouse edges is essential for people tempting to beat the house.

Besides, the operator can always by the way run contest such as follows:Anyone paying an entry fee gets a non-cashable voucher for say $1M toplay. The player must make a number of certain kinds of bets, includingsome credit ones. Every payoff will be added to the voucher. Reaching acertain winning results will grant the player a prize, which may includesome percentage of the voucher. The computer can handle contestants likeregular bettors.

Due to the fact that up to the moment of a concerning draw it doesn'tmatter when any selection is made or changed, there can be an option toallow the bettor to change selections any time before the draw. Thebettor may even purchase a ticket stating the number of certain kinds ofbets without specific selections and submit the details anytime ahead ofthe concerning draw. Computer random betting selections may also be madeavailable as an option.

There are only three similar playing surfaces with ruled movements givenhere But obviously the method can be applied to many other similarplaying surfaces with other similar ruled movements. The number of sitesand movers can easily be made different from those given. above. Ruledmovements can be different for different movers on the same playingsurface.

Other types of betting can be added into the game. Chain bets can bemore than 4 draws and other than combining mixed bits.

Thus, the scope of the invention should be determined by the appendedclaims and their legal equivalents, rather than by examples given.

1. A method to operate non-pari-mutuel movement games, which allowsplayers to bet on any one or more of millions of possible outcomes,comprising providing a plurality of moving pieces called movers on aplaying surface called site map consisting of sites, said movers andsaid site maps being displayed on monitors and printed on bet slips,providing a plurality of ruled movements, providing a plurality ofrandomly drawing said ruled movements for said movers to move on saidsite map accordingly, one round after another, and thus determiningdrawing outcomes, providing calculation of probabilities of any possiblesaid drawing outcomes, permitting players to mark said bet slips to beton any possible said drawing outcomes, whereby bet tickets will beissued, determining winning bets of said bet tickets according to saiddrawing outcomes, calculating payoff of a bet without house edge basedon bet amount and winning probability.
 2. The method of claim 1 whereinkeno bowls being used for said providing a plurality of randomlydrawing.
 3. The method of claim 1 and further comprising determininghanging bet, where a bet is defined as hanging if it stays in a positionto become winning, calculating credit for a hanging bet based on betamount and hanging probability, permitting hanging bet ticket holder toselect percentage of said credit to place bets without additionalwagering money.
 4. The method of claim 1 and further comprising:providing particular house edge functions with winning probability asvariable. using said particular house edge functions to calculate payoffof a bet based on bet amount and winning probability.
 5. The method ofclaim 1 and further comprising computerized means for displaying andmarking said bet slips.
 6. The method of claim 3 and further comprisingcomputerized means for displaying and marking said bet slips.
 7. Themethod of claim 4 and further comprising computerized means fordisplaying and marking said bet slips.